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Section 12. Physics
Rasulov Voxob Rustamovich, Researcher, of Fergana State University E-mail: [email protected] Rasulov Rustam Yavkachovich, Professor, of Fergana State University Eshboltaev I. M., Resarcher, of Kokand State Pedagogical Institute
Mamadaliyeva N. Z., Graduate student of Kokand State Pedagogical Institute.
SURFACE PHOTOCONDUCTIVITY IN A MULTIVALLEY SEMICONDUCTOR
Abstract: The surface photoconductivity in a semi-infinite multivalley semiconductor theoretically researched. It is calculated by mirror reflection of electrons o'er the surface in the approximation of the Boltzmann kinetic equation.
Keywords: the surface photoconductivity, multivalley semiconductor, mirror reflection, electron, the surface, the Boltzmann kinetic equation.
Problems of the surface today are, one of the most interests of physicists. From a purely physical point of view, the study of the surface is fundamentally important. The surface is a two-dimensional system, and not only its structure, but also many of the phenomena look on it quite differently from the volume. In essence, the surface of a solid and its "interior" are two different forms of the same substance. Therefore, surface physics has become a new field of science on the structure of matter in a condensed state.
Below we theoretically research the surface photoconductivity in a semi-infinite multivalley semiconductor. It is calculated by mirror reflection of electrons about the surface in the approximation of the Boltzmann kinetic equation.
For convenience of further calculations, we choose the following geometry of the problem. By directing the x axis in the plane zz' (z' directed along the main axis of the ellipsoidal isoenergetic surface), and the axis y is normal to this plane, we write the energy spectrum in the form e=£ Aaß va Vß or E=X Baß pa Pß , where pi
a,ß a,ß
and v{ =dE / p are the i - components of momentum and velocity, respectively.
Now let us discuss about the electron distribution function, with which the photocurrent is determined. Suppose that a light wave with frequency o is incident on a semiconductorplatewithwidth d ( - d / 2 < z < + d/ 2) perpendicular to the surface z =+d /2, and an electric field with a strength s =2 x Res exp(— at) on the current carriers and a constant external magnetic field with a strength H, where the z axis is directed along the inner normal to the surface z=+d /2, according to which, we assume that the electromagnetic wave propagates.
The nonequilibrium electron distribution function/ satisfies the kinetic equation
d + vVf - e(s + ( x Hpf = -fzfL> (!)
which is solved in the t - approximation (t - is the momentum relaxation time) by the iteration method, where it is impossible to expand f (r, t) over the spherical functions by the parameter l / |Vf /1|, because in the near-surface region/varies on the length (or 1/ot)), where lis
the mean free path of an electron, w is the frequency of the exciting light. For this reason we write the collision integral as -(f - f0)/T, where f0 - is the equilibrium distribution function of the electrons.
In the following we consider two mechanisms for the scattering of current carriers on a surface: mirror reflection (MR) and diffuse scattering (DS) with which the boundary conditions for the distribution function are determined. We assume that the energy spectrum of the carriers is spherical. Then for MR the tangential components of the momentum are conserved, and the normal component changes sign.
We will also take into account the spatial dependence of the electric field of the light wave, and also take into account that in semiconductors of cubic symmetry £z = 0.
To calculate the nonlinear electric field of the electromagnetic wave of photoconductivity (xaps and yap, where a, (3,8 = x, y, z) a semi-infinite multivalued semiconductor, we will use the energy spectrum E = Z B«P Pa pp and the calculation method proposed
a,p
in [2]. Further, we will use that when the mirror is reflected from the boundary of the surface of the semiconductor pate, we have p = p , p = pXi. And the interrelation of quantities p and pz is determined from the
law of conservation of energy, from which we have
2D 2D 2D
pZ 2 + pzl = "^T px 1 = — pxc 1 or pz 2 = - pz 1 +~Bpx 1 .
A B dE - B D
If we take into account that V =-= 2A(px--pz).
dpy B
Then Vx1 = 2B(px 1 -Dp. 1), Vx2 = Vx 1 + 2DVz 1 or
D,
A
B
Vx2 = V1 + 2 BVz1 , also px2 = pxV pz2 = pz 1 + 2tfpxV
Vx2 = Vx 1 + 2VVZ1, Vz2 = -Vz 1, где ц = D = ^.
B mxx
Consequently, on the semiconductor surface for the distribution functions f- (Vx, Vy ,Vz (o) corresponding to the current carriers coming to the surface and f+(Vx, Vy,Vz) o) corresponding to the current carriers coming from the surface of the sample, we have f-(Vx, Vy ,Vz; z = 0 ) = f+(Vx + 2nVz, Vy, - Vz; z = o),
f+(Vx, Vr,Vz;z = 0) = f-(Vx + 2nVz,Vy,-Vz;z = o). (2) We will immediately assume that the electromagnetic field is transverse, i.e. s = s(z) and the components sx, sy are nonzero. The solution of the kinetic equation for f11 = f1 (z;i) will beas
f1 = f1 (z ) = f1 (z )x exP (imt) (3)
Then
-mft +Vz df-e(l-V)df = -T (4)
d z v ' dE t Here for the electron e = |e|, F = -es . We seek the solution of the Boltzmann equation in the form
C. -fWe~'-(*-V)f*' (5)
f (z ) = e
Then
dE
я f
f (z )=Ce+Y (V )f=
yz . ,
V ге
= C1e +-ю
Y ' dE
>V ) f
(6)
г
1--
V ЮТ у
dE
Since Vz {0 the distribution function f- {vx, Vr,Vz;zz^ <»)= 0,then Cr = 0
fr(z ) =e (e-V )f
hy ' ( ' dE
r i л
(V)f (7)
1 -V
Then from condition (2) we find that for diffuse scattering (at z = 0 )
1 о 4 ' dE
sxb sin в sin ф (l + 5 sin (2ф))
с
I
\
1--
V J
-£„СCOSe
(8)
,0 fo
dE
04--^k/E
сот
The same way
fAE f Î1 - +
1 (0 dE у w
(V
f
-Y
1 - e V
v /
2
3
5
yz Vz
(9)
(1 )
The photoconductivity tensor is xaps determined by solving the Boltzmann kinetic equation [2]. In particular, for a semi-infinite multivalley semiconductor, the solution of the Boltzmann equation in the first in the electric field strength of an electromagnetic wave has the form
fjz f fi
to dE f rnr,
for mirror ref
2nsV exp(-YzV ) + £-V
(10)
ection,
x
ie d fo fl__
I )
a dE
sV|1 -exp(-YzV )
^ 2 S
+--
(i-s2)
1 exb4E exp(-yyY) )
(11)
for diffuse reflection of electrons about a surface.
The Boltzmann equation for the next iteration of the distribution function can be easily obtained in the form
f2+(0) = f2-(0) + 8~~2 nUX
V V
a z
ydE2 dE dlnEj
22 +WJ v
f &
d fo dfo d 1nT
ydE2 dE d ln E j
(12)
df df
F20(z) = -e(s0v)—0 = -es03z —0 with using of which it is possible to determine the kinetic parameters, in
dE dE _ df df particular the photoconductivity, of the sample, where F20 (z) = -e (s0v)—0 = -es03z —0.
dE dE
It is not difficult to see that in order to determine the components of the tensor xaps and Yap for a semi-infinite multivalued semiconductor, we must calculate the following integral
3 = -
e J dz | d3Vf (V )Va =-eN
or, in the case we are considering,
3 = -eN
J dz J d3Vf (V )Va
o_
J d Vfo(V )
f C2TVzVJV
„ xn 3 (m„m2 ) ?
Vz}o - 3 1 11 eN f C2rVzVad3V.
i/2
f d 3Vfo(V )
i fo(V )d 3V = -
4n( mm ))
4n (2Ef )
8yf2
Vz} o
1
e ^E.
J pp
3 (miim±m±)
Then we have the following relations for the component of the tensor xaps
X J dE
Xaxx
3
(mm!)
n
(2Ef )
e2 1 -h—e—Nn— abc I -
X o2 2 J 1
4EdE
+ 5 sin 20
sinOdOdfi
Xaxy
(miiml)
-/^e-^-rNn1 abc [ —^EdE— sinOdOd0x:VaVzr /2 c 2 J,,. 1 + ôsin2è
n
(2EF )
+ 5 sin 20
dE2 dE d ln E
VyV\.
We consider it appropriate to cestrict these results, a) in the spherical approximation of the energy spec-
because of the lack of experimental results of the spectral trum, in the sense of mutual compensation of the fluxes
dependences on photoconductivity, wich is quadratic in of electrons moving to and from the surface, the surfaces
s =(sx,£y,sz) of semi-infinite many-valley semicon- themselves are not really distinguished in any way; b) in
ductor. the case of an ellipsoidal surface, the mutual compensa-
Thus, the theory of the photoconductivity of a mul- tion of the fluxes of electrons moving to and from the
tivalley semiconductor due to the scattering of current surface, both in mirror reflection and in diffuse scatter-
carriers on the crystal surface and dependent on the de- ing, depends on the spatial variation of the distribution
gree of polarization oflight is developed. It is noted that: function and the creation of a normal electric field.
o
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